We show that a local non-Hermitian perturbation in a Hermitian lattice system generically induces scale-free localization for the continuous-spectrum eigenstates. When the perturbation lies at a finite distance to the boundary, the scale-free eigenstates are promoted to exponentially localized modes, whose number is proportional to the distance. Furthermore, when the local non-Hermitian perturbation respects parity-time $\left(\mathcal{PT}\right)$ symmetry, the $\mathcal{PT}$ symmetry breaking is always accompanied by the emergence of scale-free or exponential localization. Intriguingly, we find a concise band-structure condition which tells not only when the continuous-spectrum $\mathcal{PT}$ breaking of scale-free modes can occur but also the precise $\mathcal{PT}$-breaking energy window. Our results uncover a series of unexpected generic phenomena induced by a local non-Hermitian perturbation, which has interesting interplay with $\mathcal{PT}$ symmetry.

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• Open quantum systems & decoherence
• PT-symmetric quantum mechanics
• Topological phases of matter
• 1-dimensional systems
• Non-Hermitian systems
• Single impurity model